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Is a constant function on a measurable set is measurable?
Yes.
What are operations undo each other or the additive inverse?
Inverse operations. Additive inverse is not one operation but they are elements of a set.
What is inverse in algebra?
Given a set and a binary operation defined on the set, the inverse of any element is that element which, when combined with the first, gives the identity element for the binary operation. If the set is integers and the binary operation is addition, then the identity is 0, and the inverse of an integer k is -k. If the set is rational numbers and the binary operation is multiplication, then the identity element is 1 and the inverse of any member of the set, x (other than 0) is 1/x.
Distinguish between measurable variable and non-measurable variables of a product?
One way is to check whether the pre-image of the product is sigma-algebra. Please list an example to clarify your question.
How is the additive inverse important?
It gives closure to the set of real numbers with regard to the binary operation of addition. This makes the set a ring. The additive inverse is used, sometimes implicitly, in subtraction.
